Inclusions of von Neumann algebras and quantum groupoids II

In a previous article. in collaboration with Jean-Michel Vallin. we constructed two, quantum groupoids. dual to each other. from a depth 2 inclusion of von Neumann algebras M-0 subset of M-1. In this paper we investigate this structure in greater detail. In the previous article. we constructed the analog of a co-product. while in this paper Mr dt tini a co-inverse. by making the polar decomposition of the analog of the antipode and left and right invariant Haar operator-valued weights. These two structures of quantum groupoids dual to each other. can be placed on the relative commutants M-0' boolean AND M-2 and M-1' boolean AND M-3 in such a way that the canonical Jones' tower associated to the inclusion can be described of a tower of successive crossed-products by these two structures. (C) 2000 Academic Press.